User:Frostburn/Collatz scales: Difference between revisions

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 == Additional results ==
-Variations of the recursion do not seem work for all seeds e.g.
+Not all variations of the recursion work for all seeds e.g.
 <math>
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 </pre>
 But produces an infinite scale for <math>\frac{19}{17}</math>.
+=== Additional conjectures ===
+The recursion:
+<math>
+s_{n+1} = \frac{p_{n+1}}{q_{n+1}} = \frac{1 + 3 p_n}{4 q_n}\ \mathrm{red}\ 5
+</math>
+produces finite scales:
+<math>\left\{ s_n | n \in \mathbb{N} \right\} \cup \left\{ 5 \right\}</math>
+for all starting values <math>s_0</math>.
+Evidence
+<pre>
+! varcollatz1_3_4_5_seed_4_3.scl
+!
+Collatz scale 1, 3, 4, 5 variation of 7/5
+!
+/16
+/10
+/8
+/5
+/4
+/1
+/16
+/2
+/16
+/4
+/16
+/4
+/1
+</pre>
+The recursion:
+<math>
+s_{n+1} = \frac{p_{n+1}}{q_{n+1}} = \frac{1 + 5 p_n}{4 q_n}\ \mathrm{red}\ 3
+</math>
+produces finite scales:
+<math>\left\{ s_n | n \in \mathbb{N} \right\} \cup \left\{ 3 \right\}</math>
+for all starting values <math>s_0</math>.
+Evidence
+<pre>
+! varcollatz1_5_4_3_seed_12_7.scl
+!
+Collatz scale 1, 5, 4, 3 variation of 12/7
+!
+/336
+/6
+/336
+/2
+/7
+/336
+/1
+/28
+/48
+/56
+/4
+/24
+/1
+</pre>
+There are many (too many to list) other variations of the recursion that seem to work too, but these stood out.